Page 160 - Six Sigma Advanced Tools for Black Belts and Master Black Belts

August 31, 2006
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JWBK119-10
Monte Carlo Simulation 145
Process Capability of Case2
Calculations Based on Lognormal Distribution Model
USL
Process Data Overall Capability
LSL Pp
Target PPL
USL 75.00000 PPU 4.13
Sample Mean 19.40348 Ppk 4.13
Sample N 100 Exp. Overall Performance
Location 1.88680 PPM < LSL
Scale 0.37251 PPM > USL 0.0007565
Threshold 12.33263
PPM Total 0.0007565
Observed Performance
PPM < LSL
PPM > USL 0
PPM Total 0

16 24 32 40 48 56 64 72
Figure 10.14 Process capability study with three-parameter lognormal fit.

in Figure 10.14. We can see that the three-parameter lognormal distribution actually
fitted the data very well. The estimated C pk using this method was 4.13.

10.3.3 Comparison of results
The C pk estimated using Box--Cox transformation method was 1.25, which was 2.88
lower than that estimated by the best-fit distribution. From a Six Sigma decision point
of view, a C pk of 1.25 is not good enough and we would need to spend resources
looking into it and improving it, whereas a C pk of 4.13 is so good that we should
just leave it alone. From Figure 10.14 it is hard to believe that the USL of 75 will
be exceeded if there is no great change in the process as the specification limit is
so far (many standard deviations) away the data concentration. Therefore, from the
visual method and the reasoning given in Section 10.2.3, although both methods of
estimation are statistically acceptable, the method of using the best-fit distribution is
recommended.

10.4 MONTE CARLO SIMULATION

To further study the problem, three sets of data were artificially generated using Monte
Carlo simulation with a three-parameter lognormal distribution. The parameters were
carefully chosen so that they have an optimum λ of −3, −2 and −1 for Box--Cox trans-
formation. The process capability of each of the data sets was estimated using both
the best-fit distribution and normal approximation after Box--Cox transformation.
The Upper specification limits between 1 and 20 standard deviations away from the
mean were used to understand the differences in C pk estimation with respect to the
capability of a process.